I've always laid it out at the beginning of a game, stating that I've found that for PBP, GM rolling initiative and saves speeds things up. If there is overwhelming response that they want to roll everything, I let them. Who am I to speed up a game if they don't want it

This is fine, if those are things you declared during recruitment. However, introducing this idea mid-campaign is trickier.

I would discuss this with your players and see how they feel about it. If they like the idea, then I don't see the harm, but if even one member of the group says they want to control their own dice rolls, then I would err on the side of freedom in that one.

In my campaigns, I prefer players to handle their own rolls. It's less for me to memorize when dealing with circumstantial bonuses and it allows players to feel like they have a bit of control over the outcome, even if we all know the the dice really has the say, especially when using a dice app where you can't put the mojo on your roll. (For me, I find I get more 17s and up if I stare intently, yet lovingly, at my dice the whole time they are rolling. My dice like the attention.)

However, even though I state upfront, that I want them to post their own rolls, they will sometimes cast spells, or use skill checks without rolling for results. At which point, I just do the rolls for them and post the results in order to speed things along.

It's generally irrelevant who does the actual rolling, especially in PbP where there are no physical dice. I typically roll whatever can be rolled immediately (initiative, saves, etc) and let players roll "active" rolls relevant to their actions. The main point for me is to save back-and-forth; posting "roll initiative!" and then waiting for all players to do so, and then posting the first combat post - that's pointless and stupid. If you post the initiative to beat and allow players to act immediately if they beat it then that's not so bad because they still only need to post once (unless they lose init, in which case they are waiting anyway). I'm not really sure, though, why you wouldn't just go ahead and roll it given that you can.

I'm in favor of what tattered King said about initiative. And i think there are times that the GM should make the rolls to speed play. However, like someone else mentioned, a lot of players love to roll their own dice. Basically if you run an ad you need to be very up front about it. The people who want to roll their own dice won't apply, the people who don't care will.

To facilitate gameplay (and save a couple of days of waiting for everyone to roll initiative before they can post combat actions), I find it best if the GM rolls initiative or other reactive/passive rolls. This drastically cuts down on the back-and-forth, letting your players keep the action and story moving instead of sitting around waiting for digital dice rolls. Combat already tends to be slow enough without any extra weight added to it.

I prefer to do initiative in blocks; if PCs A-E get a 1, 2, 18, 19, and 20 respectively on their initiative, and the kobolds averaged a 12, my groups would be as follows:
Group 1: PCs C-E
Group 2: Kobolds
Group 3: PCs A and B

Within each group it doesn't matter who posts first, I adjudicate in order of posting (this speeds play IMMENSELY!). Player C could go before player E despite having an 18 initiative vs. E's 20. The players in Group 3 in this example wouldn't get to post until after the kobolds have their turn.

Block Initiative as Dalar mentioned is good, one way to improve it further (from a perspective of play speed) is by making Initiative a fixed DC based on the average initiative modifiers of the enemies. That way, if a player rolls initiative and it's over the encounter DC, they can immediately post and act without waiting for you to list out the order. There's a robust discussion of this here.

The Active Spellcasting Variant Rules are another way to put more power in the player's hands which speeds things up (we refer to them as "Spell Attack Rolls"). This, combined with publicly listed NPC armor class and Save DC's means that players can adjudicate all their own effects right away, rather than having to go back and forth. We leave PC Saving Throws in the hand of the player, which is almost always fine, since they can just roll them before they act in the next round and it usually doesn't impact too many things to wait for the player to do so. Metagaming is a moderate concern.

As a player I prefer to roll my own attacks. For the one game I DM for I gave the players the option to roll their own initiative or have me do it. As for battle I just tell the players to post their action no matter what order they are in and I will sort it out. If a target has been killed I will check with a player if it totally disrupts their action and movement but generally just move whatever attack they have to the next target.

This way all players can post instead of waiting for the player ahead of them in battle order.